Finite Group Databasebeta

← #13144037

Subgroups — #13144037

Order 1728.

836 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722675C211yesmaximal, characteristic
2#11721872C211yesmaximal, characteristic, complemented
3#11721788C211yesmaximal, characteristic, complemented
4#11721721C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
6#11720560C211yesmaximal, characteristic, complemented
7#11720522C211yesmaximal, characteristic, complemented
8C3×#4825—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10#88748—21no—
11D4×#272—22no—
12D4×#272—21no—
13#88740—21no—
14#88740C2211yescharacteristic
15C2×C4⋊#272—21no—
16C2×C4⋊#272C2211yescharacteristic
17C2×#6229C2211yescharacteristic, complemented
18M16×H(3,𝔽3)C2211yescharacteristic, complemented
19C42×H(3,𝔽3)C2211yescharacteristic, complemented
20#88550—21no—
21#88546—21no—
22#88542—22no—
23#88538C2211yescharacteristic, complemented
24#88530—21no—
25#88528—21no—
26C3×#1009—34no—
27C3×#951—34no—
28C3×#909—34no—
29C3×#875—34no—
30C32×#104D314yescomplemented
31C3×#823—34no—
32C3×#805—34no—
33C22×#272—22no—
34C22×#272D412yes—
35Q8×H(3,𝔽3)D411yescharacteristic
36D4×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#6235—22no—
42#6235—21no—
43C2×#1084—21no—
44C4⋊#272—22no—
45C4⋊#272—21no—
46C4⋊#272—21no—
47C4⋊#272D411yescharacteristic
48C4×#272—22no—
49C4×#272—21no—
50#6229—21no—
51#6229—21no—
52#6229D411yescharacteristic
53C8×H(3,𝔽3)—21no—
54#6180—21no—
55C3×#455—91no—
56#4825—34no—
57C32×#41C2×D314yes—
58C3×#239—64no—
59C3×D3×D4—68no—
60C3×D3×D4—64no—
61C3×#235—34no—
62C3×#235—64no—
63C2×C3×D12—34no—
64C2×C3×D12—64no—
65C2×C3×Dic6—34no—
66C32×M16C2×D314yescomplemented
67C32×C42C2×D314yescomplemented
68C3×#216—64no—
69C3×#214—64no—
70C3×(C2×D3)⋊C4—68no—
71C3×#208—34no—
72C3×Dic12—64no—
73C3×#204—64no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C2×#272—22no—
77C2×#272—21no—
78C2×#272—42no—
79C2×#272—82no—
80C4×H(3,𝔽3)—21no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)C2×D411yescharacteristic
83C4×H(3,𝔽3)C2×D411yescharacteristic
84#1084—21no—
85#1084—41no—
86C3×H(5,𝔽2)—91no—
87#1009—34no—
88C3×#137—91no—
89C3×#137—91no—
90C3×#124—91no—
91#951—34no—
92#909—34no—
93#875—34no—
94C3×#104—34no—
95C3×#104—91no—
96C3×#104C3⋊D311yescharacteristic
97C3×#100—91no—
98#823—34no—
99#805—34no—
100C22×C3×D3—38no—
101C22×C3×D3—68no—
102C32×Q8D1214yes—
103C32×D4—24no—
104C32×D4D1214yes—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4C22×D314yes—
108C3×#67—68no—
109C3×#67—64no—
110C2×C3×Dic3—64no—
111C3×D12—34no—
112C3×D12—68no—
113C3×D12—64no—
114C3×D12—64no—
115C3×C4×D3—68no—
116C3×C4×D3—64no—
117C3×Dic6—34no—
118C3×Dic6—64no—
119C3×Dic6—64no—
120C32×C8—24no—
121C3×C3⋊C8—64no—
122#455—91noSylow
123C2×H(3,𝔽3)—21no—
124C2×H(3,𝔽3)—41no—
125C2×H(3,𝔽3)#12011yescharacteristic
126#272—42no—
127#272—41no—
128C3×#41—34no—
129C3×#41—91no—
130C3×#41—181no—
131C3×#41—181no—
132C3×#41C2×C3⋊D311yescharacteristic
133C2×C3×Q8—91no—
134C2×C3×D4—91no—
135C2×C3×D4—181no—
136C2×C3×D4—182no—
137C2×C3×D4—181no—
138#239—64no—
139D3×D4—64no—
140D3×D4—68no—
141#235—34no—
142#235—64no—
143C2×D12—34no—
144C2×D12—64no—
145C2×Dic6—34no—
146C3×Q16—181no—
147C3×Q16—181no—
148C3×SD16—181no—
149C3×SD16—181no—
150C3×M16—34no—
151C3×M16—91no—
152C3×M16C2×C3⋊D311yescharacteristic
153C3×C22⋊C4—182no—
154C3×C42—34no—
155C3×C42C2×C3⋊D311yescharacteristic
156#216—64no—
157#214—64no—
158(C2×D3)⋊C4—68no—
159#208—34no—
160Dic12—64no—
161#204—64no—
162C22×C32—24no—
163C22×C32D3×D414yes—
164C2×C3×D3—64no—
165C2×C3×D3—68no—
166C2×C3×D3—128no—
167C2×C3×D3—248no—
168C32×C4—24no—
169C32×C4—44no—
170C32×C4C2×D1214yes—
171C32×C4D3×D414yes—
172C3×Dic3—64no—
173C3×Dic3—124no—
174H(5,𝔽2)—91no—
175#137—91no—
176#137—91no—
177#124—91no—
178#104—91no—
179#104#27211yescharacteristic, complemented
180#100—91no—
181H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
182C23×C3—92no—
183C23×C3—182no—
184C22×D3—38no—
185C22×D3—68no—
186C3×Q8—34no—
187C3×Q8—91no—
188C3×Q8—181no—
189C3×Q8—181no—
190C3×Q8C4⋊(C3⋊D3)11yescharacteristic
191C3×D4—21no—
192C3×D4—34no—
193C3×D4—64no—
194C3×D4—91no—
195C3×D4—182no—
196C3×D4—182no—
197C3×D4—181no—
198C3×D4—181no—
199C3×D4—181no—
200C3×D4C4⋊(C3⋊D3)11yescharacteristic
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—34no—
204C2×C3×C4—64no—
205C2×C3×C4—64no—
206C2×C3×C4—182no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4C22×C3⋊D311yescharacteristic
210#67—64no—
211#67—68no—
212C2×Dic3—64no—
213D12—34no—
214D12—64no—
215D12—64no—
216D12—68no—
217C4×D3—64no—
218C4×D3—68no—
219Dic6—34no—
220Dic6—64no—
221Dic6—64no—
222C3×C8—21no—
223C3×C8—64no—
224C3×C8—181no—
225C3⋊C8—64no—
226C2×C32—24no—
227C2×C32—44no—
228C2×C32#88214yes—
229C3×D3—128no—
230C3×D3—124no—
231#41—91no—
232#41—181no—
233#41—181no—
234#41C2×#27211yescharacteristic
235C2×Q8—91no—
236C2×D4—91no—
237C2×D4—181no—
238C2×D4—182no—
239C2×D4—181no—
240Q16—181no—
241Q16—181no—
242SD16—181no—
243SD16—181no—
244M16—91no—
245M16C2×#27211yescharacteristic, complemented
246C22⋊C4—182no—
247C42C2×#27211yescharacteristic, complemented
248C22×C3—21no—
249C22×C3—34no—
250C22×C3—64no—
251C22×C3—181no—
252C22×C3—182no—
253C22×C3—362no—
254C22×C3—722no—
255C22×C3D4×C3⋊D311yescharacteristic
256C2×D3—64no—
257C2×D3—68no—
258C2×D3—128no—
259C2×D3—248no—
260C3×C4—21no—
261C3×C4—34no—
262C3×C4—34no—
263C3×C4—41no—
264C3×C4—64no—
265C3×C4—124no—
266C3×C4—181no—
267C3×C4—361no—
268C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
269C3×C4D4×C3⋊D311yescharacteristic
270Dic3—64no—
271Dic3—124no—
272C32#482514yescomplemented
273C23—92no—
274C23—182no—
275Q8—91no—
276Q8—181no—
277Q8—181no—
278Q8C4⋊#27211yescharacteristic
279D4—21no—
280D4—91no—
281D4—182no—
282D4—182no—
283D4—181no—
284D4—181no—
285D4—181no—
286D4C4⋊#27211yescharacteristic
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—182no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C4C22×#27211yescharacteristic
293C8—21no—
294C8—181no—
295C2×C3—21no—
296C2×C3—34no—
297C2×C3—41no—
298C2×C3—64no—
299C2×C3—124no—
300C2×C3—362no—
301C2×C3—361no—
302C2×C3#6409111yescharacteristic
303D3—128no—
304D3—124no—
305C22—21no—
306C22—181no—
307C22—182no—
308C22—362no—
309C22—722no—
310C22D4×#27211yescharacteristic
311C4—21no—
312C4—41no—
313C4—181no—
314C4—361no—
315C4C2×C4⋊#27211yescharacteristic
316C4D4×#27211yescharacteristic
317C3—34no—
318C3#1059252411yescharacteristic
319C2—21no—
320C2—41no—
321C2—362no—
322C2—361no—
323C2#1172173811yescharacteristic
subgroupG/H
center, socleC2×C3#64091
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#104C2
prime radical#11722675C2
hypercenterC3×#104C3⋊D3
hypocenterH(3,𝔽3)#455
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#104

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11721872C2semidirect
3. #11721788C2semidirect
4. #11721721C2semidirect
5. H(3,𝔽3)×#104C2semidirect
6. #11720560C2semidirect
7. #11720522C2semidirect
17. C2×#6229C22semidirect
18. M16×H(3,𝔽3)C22semidirect
19. C42×H(3,𝔽3)C22semidirect
23. #88538C22semidirect
30. C32×#104D3semidirect
66. C32×M16C2×D3semidirect
67. C32×C42C2×D3semidirect
179. #104#272semidirect
181. H(3,𝔽3)#455semidirect
245. M16C2×#272semidirect
247. C42C2×#272semidirect
272. C32#4825semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117226752
#117218722
#117217882
#117217212
H(3,𝔽3)×#1042
#117205602
#117205222
C3×#48253